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Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

Mauricio Che, Fernando Galaz-García, Martin Kerin, Jaime Santos-Rodríguez

Abstract

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to $\mathbb{P}_2$. We then prove that infinite rays are isometrically rigid with respect to $\mathbb{P}_p$ for any $p\geq 1$, whereas taking infinite half-cylinders (i.e.\ product spaces of the form $X\times [0,\infty)$) over compact non-branching geodesic spaces preserves isometric rigidity with respect to $\mathbb{P}_p$, for $p>1$. Finally, we prove that spherical suspensions over compact spaces with diameter less than $π/2$ are isometrically rigid with respect to $\mathbb{P}_p$, for $p>1$.

Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

Abstract

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the -Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to . We then prove that infinite rays are isometrically rigid with respect to for any , whereas taking infinite half-cylinders (i.e.\ product spaces of the form ) over compact non-branching geodesic spaces preserves isometric rigidity with respect to , for . Finally, we prove that spherical suspensions over compact spaces with diameter less than are isometrically rigid with respect to , for .

Paper Structure

This paper contains 15 sections, 44 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wasserstein spaces and optimal transport
  4. Geodesics, midpoints and non-branching condition
  5. Proof of theorem \ref{['t:flexibility of spaces with lines']}: Non-rigidity of spaces splitting a Hilbert space
  6. Proof of theorem \ref{['t:rigidity of rays']}: Isometric rigidity of rays
  7. Proof of theorem \ref{['t:rigidity of cylinders']}: Isometric rigidity of half-cylinders
  8. Invariance of measures on the base of the half-cylinder
  9. Invariance of Dirac measures
  10. Invariance of atomic measures
  11. Proof of theorem \ref{['t:rigidity of compact spaces']}: Isometric rigidity of spherical suspensions
  12. Invariance of Dirac measures
  13. Invariance of measures on meridians
  14. Invariance of measures on parallels
  15. Isometric rigidity of suspensions.

Key Result

Theorem 1

Let $(Y, d_Y)$ be a proper metric space and $(H,|\cdot|)$ be a separable Hilbert space with $\dim H\geq 1$. Let the product $X = H \times Y$ be equipped with the product metric Then $(\mathbb{P}_2(X), \mathrm{W}_2)$ admits an isometric action of the orthogonal group of $H$ by exotic isometries.

Figures (2)

  • Figure 1: In this diagram, the deep grey regions represent $\mathop{\mathrm{supp}}\nolimits(\nu_0)\cap X\times\{0,2D\}$; the dark regions represent $\mathop{\mathrm{supp}}\nolimits(\nu_1)\cap X\times \{0,2D\}$; and pale grey region on top represent $\mathop{\mathrm{supp}}\nolimits(\nu_0)\cap X\times\{4D\}=\mathop{\mathrm{supp}}\nolimits(\nu_1)\cap X\times\{4D\}$. The arrows represent any optimal plan between $\nu_0$ and $\nu_1$; any such optimal plan fixes the mass on $X\times \{4D\}$.
  • Figure 2: In this diagram, $\mu$ is supported on the black dots, $\widetilde{\mu}$ is supported on the grey ones, and $\nu$ is supported on the endpoints of the suspension, $\mathbf{0}$ and $\boldsymbol{\pi}$, in such a way that $\mu\in\mathop{\mathrm{Int}}\nolimits(\widetilde{\mu},\nu)$. Any other measure $\mu'\in\mathop{\mathrm{Int}}\nolimits(\widetilde{\mu},\nu)$ is atomic, and only differs from $\mu$ by the mass it gives to the points $\{[x_{i1},t_1]\}_{i=1}^{k_1}$, $\{[x_{i1},t_2]\}_{i=1}^{k_1}$, $\{[x_{i2},t_1]\}_{i=1}^{k_2}$, and $\{[x_{i2},t_2]\}_{i=1}^{k_2}$, i.e. $\mu'$ may give positive mass to the white dots.

Theorems & Definitions (96)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 86 more